244 research outputs found
Tensor structure from scalar Feynman matroids
We show how to interpret the scalar Feynman integrals which appear when
reducing tensor integrals as scalar Feynman integrals coming from certain nice
matroids.Comment: 12 pages, corrections suggested by referee
Advances on Strictly -Modular IPs
There has been significant work recently on integer programs (IPs)
with a constraint marix
with bounded subdeterminants. This is motivated by a well-known conjecture
claiming that, for any constant , -modular
IPs are efficiently solvable, which are IPs where the constraint matrix has full column rank and all minors of
are within . Previous progress on this question, in
particular for , relies on algorithms that solve an important special
case, namely strictly -modular IPs, which further restrict the minors of to be within . Even for ,
such problems include well-known combinatorial optimization problems like the
minimum odd/even cut problem. The conjecture remains open even for strictly
-modular IPs. Prior advances were restricted to prime , which
allows for employing strong number-theoretic results.
In this work, we make first progress beyond the prime case by presenting
techniques not relying on such strong number-theoretic prime results. In
particular, our approach implies that there is a randomized algorithm to check
feasibility of strictly -modular IPs in strongly polynomial time if
Certificates and relaxations for integer programming and the semi-group membership problem
We consider integer programming and the semi-group membership problem. We
provide the following theorem of the alternative: the system Ax=b has no
nonnegative integral solution x if and only if p(b) <0 for some given
polynomial p whose vector of coefficients lies in a convex cone that we
characterize. We also provide a hierarchy of linear programming relaxations,
where the continuous case Ax=b with x real and nonnegative, describes the first
relaxation in the hierarchy.Comment: 21 page
On Augmentation Algorithms for Linear and Integer-Linear Programming: From Edmonds-Karp to Bland and Beyond
Motivated by Bland's linear-programming generalization of the renowned
Edmonds-Karp efficient refinement of the Ford-Fulkerson maximum-flow algorithm,
we discuss three closely-related natural augmentation rules for linear and
integer-linear optimization. In several nice situations, we show that
polynomially-many augmentation steps suffice to reach an optimum. In
particular, when using "discrete steepest-descent augmentations" (i.e.,
directions with the best ratio of cost improvement per unit 1-norm length), we
show that the number of augmentation steps is bounded by the number of elements
in the Graver basis of the problem matrix, giving the first ever strongly
polynomial-time algorithm for -fold integer-linear optimization. Our results
also improve on what is known for such algorithms in the context of linear
optimization (e.g., generalizing the bounds of Kitahara and Mizuno for the
number of steps in the simplex method) and are closely related to research on
the diameters of polytopes and the search for a strongly polynomial-time
simplex or augmentation algorithm
Absolute Objects and Counterexamples: Jones-Geroch Dust, Torretti Constant Curvature, Tetrad-Spinor, and Scalar Density
James L. Anderson analyzed the novelty of Einstein's theory of gravity as its
lack of "absolute objects." Michael Friedman's related work has been criticized
by Roger Jones and Robert Geroch for implausibly admitting as absolute the
timelike 4-velocity field of dust in cosmological models in Einstein's theory.
Using the Rosen-Sorkin Lagrange multiplier trick, I complete Anna Maidens's
argument that the problem is not solved by prohibiting variation of absolute
objects in an action principle. Recalling Anderson's proscription of
"irrelevant" variables, I generalize that proscription to locally irrelevant
variables that do no work in some places in some models. This move vindicates
Friedman's intuitions and removes the Jones-Geroch counterexample: some regions
of some models of gravity with dust are dust-free and so naturally lack a
timelike 4-velocity, so diffeomorphic equivalence to (1,0,0,0) is spoiled.
Torretti's example involving constant curvature spaces is shown to have an
absolute object on Anderson's analysis, viz., the conformal spatial metric
density. The previously neglected threat of an absolute object from an
orthonormal tetrad used for coupling spinors to gravity appears resolvable by
eliminating irrelevant fields. However, given Anderson's definition, GTR itself
has an absolute object (as Robert Geroch has observed recently): a change of
variables to a conformal metric density and a scalar density shows that the
latter is absolute.Comment: Minor editing, small content additions, added references. Forthcoming
in_Studies in History and Philosophy of Modern Physics_, June 200
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